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 {\Large \bf Article reference: JPhysA-103832}

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 {\Large \sf Separable Decomposition and Quantum Correlations in  Toeplitz Matrices}\\
   {\large \sf by Marek Mozrzymas, Adam Rutkowski  and Michal Studzinski}


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The manuscript by Mozrzymas et al. (reference: JPhysA-103832) investigates  the separability conditions in the case of block-Toeplitz matrices and
unfortunately  the outcomes  do not significantly advance the field of the characterization of quantum correlations in multi-qubit systems. The second part (section III), devoted the quantum discord existing in quantum states having the Toeplitz  structure, seems to be heavily dependent on the paper "Circulant States with Vanishing Quantum Discord" by B. Bylicka et al (Open Syst. Inf. Dyn. 19, 1250006 (2012)). The paper is essentially a reformulation of existing results.
I do not feel that there is enough relevant material. This is the crucial reason behind my negative assessment and  I can not recommend the paper for publication. On the other hand, the authors explain that Toeplitz matrices play an important role in several mathematical and physical fields
and mention some applications. However, the generation of quantum states of Toeplitz type seems to be
very difficult and I don't know what is the best efficiency scheme to generate them with the present day technology. 



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I do not recommend publication of the paper as it is.

The paper is definitely correct and is a nice analysis, but I have
some concerns. It seems to be heavily dependent on the paper
(or papers) by Karabali and Nair. The latter authors presented a
general group-theoretic scheme for analyzing the Landau problem
on various coset manifolds. They also showed how one could
construct the $\nu =1$ Hall state, which is incompressible as well.
Further they analyzed the edge state effective theories for ${\bf CP}^k$.
The present authors have also written papers extending their analysis
to Bargmann ball $B^k$.

In this paper, the authors do a similar analysis for the coset space
$SU(3)/U(1)\times U(1)$. This case is a straightforward extension
of previous analyses. The results are of some interest, but, in the
light of the work by KN and the previous work by the authors
themselves, it is difficult to say that these papers break new ground.
Even so, if the results had any dramatic observational consequence,
I would recommend publication. But as it is, the result is, primarily,
of mathematical interest. The authors indicate at the end that they
plan to look at higher flag manifolds. So my recommendation is that
they should carry this out and the totality of the work would be
perhaps more appropriate and adequate for publication. 






REFEREE REPORTS:
Referee: 1

Recommendation: Reject (average work that adds little new knowledge to the field)

Comments:
In my opinion, the main results of this work are simple reformulations of the separability conditions in the case of block-Toeplitz matrices. As such, they appear evident to me and not worth publishing.

More specifically:

1. Theorem 1  (page 4 of the manuscript) assumes that all blocks of the block-Toeplitz matrix commute. Then it is obvious to diagonalize all of them simultaneously and re-write the original matrix in the corresponding basis. Then it will separate into a sum of Toeplitz matrices. The theorem states that if those Toeplitz matrices are all positive semidefinite, then this decomposition is the same as in the definition of a separable state. This appears to me a trivial application of the separability definition.

2. Remark 3 (page 7 of the manuscript) gives a condition for the absence of quantum discord, which is a trivial application of the Theorem 2 (borrowed from ref [14]). The condition (40) mentioned in Remark 3 is simply a way to write the condition that matrices involved in the decomposition commute, as required by Theorem 2.

In my understanding, theorem 1 and remark 3 are supposed to be the main results of the paper. Both of them appear trivial to me (modulo Theorem 2, which is not trivial, but whose proof is not part of this paper). So, while the paper seems to be correct, I think it adds little knowledge to the field and should not be published.

CONFIDENTIAL COMMENTS TO THE EDITOR(S):
I have certain expertise in Toeplitz matrices and quantum entenglement, but I have little expertise in the theory of separable states. My negative report is based on the analysis of the proofs presented in the paper, which I find simple reformulations of definitions, together with previously existing results. It may be worth asking for an independent opinion from an expert specifically in the theory of separable states.

Additional Questions:
Originality: 5
Scientific Rigour: 10
Significance: 2

What impact will this article have in the field?: Low

Has the author made reference to the most recent and most appropriate work? <br>(If no, please provide references in the Comments to the author(s) section.): Yes

Is the article well written and the work clearly communicated? <br>(Please note we will edit upon acceptance and correct any grammatical or typographical errors.): Yes

Is this article better suited to a more specialised journal?<br>(If yes, please indicate in the Confidential comments to the editor(s) section which journal it would be suitable for.): No

Do you think your quality assessment scores will change if the author satisfactorily implements your recommendations?<br>(If no, please explain why in the Confidential comments to the editor(s) section.): No

Is this article fully or partially in your area of expertise? <br>(If partially, please provide an explanation in the Confidential comments to the editor(s) section.): Partially in my area of expertise


Referee: 2

Recommendation: Accept

Comments:
This is an interesting work that links entanglement and quantum discord to Toeplitz matrices. By this the authors succeeded to construct explicitly separable decompositions which is a nontrivial task. The work is also well motivated and show interesting interdisciplinary results. Therefore I suggest the publication. However, there are extremely many typos (please, consider to use immediately spelling control within your LateX interpreter!!) and the English should be corrected (often articles are missing). See some (only few) examples below. The results presented in the article are enough to be considered as a publication, however, the authors may think about all the other types of entanglement such as multipartite entanglement, whether also such a connection can be found.

1.)Abstract: genreral ->general
2.) many times: symplex  ->simplex
3.) Page 5: propieties
4.) Line 50: In this section ...quantum discord BEHAVES WITHIN the class of Toeplitz matrices  STUDIED ...
5.) Line 55: let us remind here --> let us recall the  theorem ...
6.)Line 58: we are given with --> we have given
7.) Page 6:Line9: writtien
8.)  Line 16: thogeter
9.) Line 19: liketo
...
10.) References [7]: comma missing, sometimes the Journals are written out sometimes not,...
...

CONFIDENTIAL COMMENTS TO THE EDITORI(S):
I am an expert in entanglement theory and information science which this works links to but not on the Toeplitz matrices. I found the connection presented exciting, however, there English needs correction and there were many typos.

Additional Questions:
Originality: 7
Scientific Rigour: 8
Significance: 7

What impact will this article have in the field?: Medium

Has the author made reference to the most recent and most appropriate work? <br>(If no, please provide references in the Comments to the author(s) section.): Yes

Is the article well written and the work clearly communicated? <br>(Please note we will edit upon acceptance and correct any grammatical or typographical errors.): Yes

Is this article better suited to a more specialised journal?<br>(If yes, please indicate in the Confidential comments to the editor(s) section which journal it would be suitable for.): No

Do you think your quality assessment scores will change if the author satisfactorily implements your recommendations?<br>(If no, please explain why in the Confidential comments to the editor(s) section.): Yes

Is this article fully or partially in your area of expertise? <br>(If partially, please provide an explanation in the Confidential comments to the editor(s) section.): Partially in my area of expertise

Letter reference: IS10

